Define free charge and bound charge; express bound charges in terms of polarization P.

Define free charge and bound charge; express bound charges in terms of polarization P.

• A. ρ_b = −∇ · P; σ_b = P · n̂.
• B. ρ_b = ∇ × P; σ_b = P × n̂.
• C. ρ_b = ∇ · P; σ_b = −P · n̂.
• D. ρ_f = ∇ · P; σ_f = P · n̂.

Answer: (A)

When a dielectric becomes polarized, bound charges appear because the dipoles inside the material create a distribution of charge that is attached to the medium. The volume bound charge density is given by ρ_b = −∇·P. The minus sign comes from how the polarization P (dipole moment per unit volume) sources or sinks charge: if the polarization diverges outward, it corresponds to negative bound charge, and if it converges, positive bound charge. This divergence of P captures how the polarization changes from point to point inside the material. At the surface of the dielectric, bound charge accumulates where the polarization ends, and this is described by the surface bound charge density σ_b = P · n̂, where n̂ is the outward normal to the surface. The normal component of P tells you how much dipole moment is leaving the material at the boundary, which manifests as surface charge. Free charge density ρ_f, on the other hand, is not determined by P. It represents charges not bound to the material, and relations like ρ_f = ∇·P or σ_f = P · n̂ would mix concepts incorrectly. The correct expressions for bound charges are the ones above.

When a dielectric becomes polarized, bound charges appear because the dipoles inside the material create a distribution of charge that is attached to the medium. The volume bound charge density is given by ρ_b = −∇·P. The minus sign comes from how the polarization P (dipole moment per unit volume) sources or sinks charge: if the polarization diverges outward, it corresponds to negative bound charge, and if it converges, positive bound charge. This divergence of P captures how the polarization changes from point to point inside the material.

At the surface of the dielectric, bound charge accumulates where the polarization ends, and this is described by the surface bound charge density σ_b = P · n̂, where n̂ is the outward normal to the surface. The normal component of P tells you how much dipole moment is leaving the material at the boundary, which manifests as surface charge.

Free charge density ρ_f, on the other hand, is not determined by P. It represents charges not bound to the material, and relations like ρ_f = ∇·P or σ_f = P · n̂ would mix concepts incorrectly. The correct expressions for bound charges are the ones above.